24576/24565: Difference between revisions
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Comma list: {{monzo| 13 1 -3 }} = 24576/24565 | Comma list: {{monzo| 13 1 -3 }} = 24576/24565 | ||
{{mapping|legend=1| 1 5 6 | 0 3 1 }} | |||
[[CTE]] generator: 85/64 = 491.541{{cent}} | |||
{{Optimal ET sequence|legend=1| 5, 17, 22, 61, 83 }} | {{Optimal ET sequence|legend=1| 5, 17, 22, 61, 83 }} | ||
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Subgroup: 2.75.85.9/7 | Subgroup: 2.75.85.9/7 | ||
Comma list: {{monzo| 13 1 -3 0 }} = 24576/24565, {{monzo| 2 -2 0 1 }} = 2025/2023 | Comma list: {{monzo| 13 1 -3 0 }} = 24576/24565, {{monzo| 2 -2 0 1 }} = 2025/2023 | ||
Some good (relative to their size) EDOs supporting it: 5, 12, 17, 22, 27, 39, 49, 61, 71, 83, 105, 127, 149, 159, 171 | Some good (relative to their size) EDOs supporting it: 5, 12, 17, 22, 27, 39, 49, 61, 71, 83, 105, 127, 149, 159, 171 | ||
{{mapping|legend=1| 1 5 6 2 | 0 3 1 -4 }} | |||
[[CTE]] generator: 85/64 = 491.338 | |||
It should be noted that just because these are good for the generators given that does not mean that they are good for the broader 2.3.5.7.17 subgroup, so one may need to take supersets in that case, in which case again it is preferred to look at the next extension. | It should be noted that just because these are good for the generators given that does not mean that they are good for the broader 2.3.5.7.17 subgroup, so one may need to take supersets in that case, in which case again it is preferred to look at the next extension. | ||
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We may observe that in a good tuning of archagall there is an accurate [[5/4]] at +13 fourths ([[85/64]]'s) minus five octaves ([[2/1]]'s). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to [[171edo]] for which 171edo is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = [[1225/1224]] and (S18/S20)/S49 = [[5832/5831]] while not tempering any of {S49, S50, S18/S20} individually. Note that 171edo is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it. | We may observe that in a good tuning of archagall there is an accurate [[5/4]] at +13 fourths ([[85/64]]'s) minus five octaves ([[2/1]]'s). Because 75/25 = 3 and 85/5 = 17 this allows us to collapse it into its corresponding prime subgroup. This temperament is very closely related to [[171edo]] for which 171edo is the tuning tempering {S49, S50, S18/S20} which is natural because this temperament tempers S49*S50 = S35 = [[1225/1224]] and (S18/S20)/S49 = [[5832/5831]] while not tempering any of {S49, S50, S18/S20} individually. Note that 171edo is exceptionally efficient and accurate in the 2.3.5.7.17 subgroup, constituting a microtemperament for it. | ||
Subgroup: 2.3.5.7.17 | [[Subgroup]]: 2.3.5.7.17 | ||
Comma list: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35 | Comma list: 24576/24565 = S16/S17, 57375/57344 = S15/S16, 1225/1224 = S35 | ||
{{mapping|legend=1| 1 11 -3 20 9 | 0 -23 13 -42 -12 }} | |||
[[CTE]] generator: 85/64 = 491.222{{cent}} | |||
Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364 | Some good (relative to their size) EDOs supporting it: 22, 149, 171, 193, 215, 320, 364 | ||
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Subgroup: 2.3.5.17 | Subgroup: 2.3.5.17 | ||
{{mapping|legend=1| 1 1 2 4 | 0 1 1 0 | 0 0 -3 1 }} | |||
[[CTE]] generators: (2/1,) 3/2 = 701.943, 17/16 = 105.201 | |||
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }} | {{Optimal ET sequence|legend=1| 10, 12, 22, 34, 80, 103, 115, 125, 137, 159, 171, 354, 376, 388, 559, 1882, 2441g, 3000g, 6559gg, 9559cggg }} | ||
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[[Category:Mavka]] | [[Category:Mavka]] | ||
[[Category:Commas with unknown etymology]] | |||